Zeta and Related Functions - 25.14 Lerch’s Transcendent

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
25.14.E2	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\Phi\left(1,s,a\right)}}$ \Hurwitzzeta@{s}{a} = \LerchPhi@{1}{s}{a}	${\displaystyle{\displaystyle\Re s>1}}$	Zeta(0, s, a) = LerchPhi(1, s, a)	HurwitzZeta[s, a] == LerchPhi[1, s, a]	Successful	Failure	-	Successful [Tested: 2]
25.14.E3	${\displaystyle{\displaystyle\mathrm{Li}_{s}\left(z\right)=z\Phi\left(z,s,1\right)}}$ \polylog{s}@{z} = z\LerchPhi@{z}{s}{1}	${\displaystyle{\displaystyle\Re s>1,|z|\leq 1}}$	polylog(s, z) = z*LerchPhi(z, s, 1)	PolyLog[s, z] == z*LerchPhi[z, s, 1]	Successful	Successful	-	Successful [Tested: 10]
25.14.E4	${\displaystyle{\displaystyle\Phi\left(z,s,a\right)=z^{m}\Phi\left(z,s,a+m\right)+\sum_{n=0}^{m-1}\frac{z^{n}}{(a+n)^{s}}}}$ \LerchPhi@{z}{s}{a} = z^{m}\LerchPhi@{z}{s}{a+m}+\sum_{n=0}^{m-1}\frac{z^{n}}{(a+n)^{s}}		LerchPhi(z, s, a) = (z)^(m)* LerchPhi(z, s, a + m)+ sum(((z)^(n))/((a + n)^(s)), n = 0..m - 1)	LerchPhi[z, s, a] == (z)^(m)* LerchPhi[z, s, a + m]+ Sum[Divide[(z)^(n),(a + n)^(s)], {n, 0, m - 1}, GenerateConditions->None]	Failure	Successful	Failed [6 / 300] Result: .27656730e-2-.27656730e-2*I Test Values: {a = -3/2, s = -2, z = 1/2*3^(1/2)+1/2*I, m = 2} Result: 0.+.228647547e-1*I Test Values: {a = -3/2, s = -2, z = -1/2+1/2*I*3^(1/2), m = 2} ... skip entries to safe data	Successful [Tested: 300]
25.14.E5	${\displaystyle{\displaystyle\Phi\left(z,s,a\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1-ze^{-x}}\mathrm{d}x}}$ \LerchPhi@{z}{s}{a} = \frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1-ze^{-x}}\diff{x}	${\displaystyle{\displaystyle\Re s>0,\Re a>0}}$	LerchPhi(x + y*I, s, a) = (1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(1 -(x + y*I)*exp(- x)), x = 0..infinity)	LerchPhi[x + y*I, s, a] == Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],1 -(x + y*I)*Exp[- x]], {x, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Failed [162 / 162] Result: Plus[Complex[0.29818646299224294, -0.45270555517796296], Times[-1.1283791670955126, NIntegrate[Complex[0.15484016278663867, -0.07789552790412994] Test Values: {1.5, 0, DirectedInfinity[1]}]]], {Rule[a, 1.5], Rule[s, 1.5], Rule[x, 1.5], Rule[y, -1.5], Rule[z, Rational[1, 2]]} Result: Plus[Complex[0.29818646299224244, 0.45270555517796246], Times[-1.1283791670955126, NIntegrate[Complex[0.15484016278663867, 0.07789552790412994] Test Values: {1.5, 0, DirectedInfinity[1]}]]], {Rule[a, 1.5], Rule[s, 1.5], Rule[x, 1.5], Rule[y, 1.5], Rule[z, Rational[1, 2]]} ... skip entries to safe data
25.14.E6	${\displaystyle{\displaystyle\Phi\left(z,s,a\right)=\frac{1}{2}a^{-s}+\int_{0}^{\infty}\frac{z^{x}}{(a+x)^{s}}\mathrm{d}x-2\int_{0}^{\infty}\frac{\sin\left(x\ln z-s\operatorname{arctan}\left(x/a\right)\right)}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\mathrm{d}x}}$ \LerchPhi@{z}{s}{a} = \frac{1}{2}a^{-s}+\int_{0}^{\infty}\frac{z^{x}}{(a+x)^{s}}\diff{x}-2\int_{0}^{\infty}\frac{\sin@{x\ln@@{z}-s\atan@{x/a}}}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\diff{x}	${\displaystyle{\displaystyle\Re s>0,|z|<1,\Re s>1,|z|=1,\Re a>0}}$	LerchPhi(x + y*I, s, a) = (1)/(2)*(a)^(- s)+ int(((x + y*I)^(x))/((a + x)^(s)), x = 0..infinity)- 2*int((sin(x*ln(x + y*I)- s*arctan(x/a)))/(((a)^(2)+ (x)^(2))^(s/2)*(exp(2*Pi*x)- 1)), x = 0..infinity)	LerchPhi[x + y*I, s, a] == Divide[1,2]*(a)^(- s)+ Integrate[Divide[(x + y*I)^(x),(a + x)^(s)], {x, 0, Infinity}, GenerateConditions->None]- 2*Integrate[Divide[Sin[x*Log[x + y*I]- s*ArcTan[x/a]],((a)^(2)+ (x)^(2))^(s/2)*(Exp[2*Pi*x]- 1)], {x, 0, Infinity}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out